Question

1. Approximately 43.2% of patients are female. In a random selection of 10 patients from the full 95 patients, calculate the following probabilities. Use the binomial distribution with parameters n=10 and p=0.432. Show all work.
   1. probability at least 1 patient is a female in the 10 selected, P(X≥1) (Hint: use the rule of complements)
   2. probability that more than 3 but less than 7 patients are female, P(3<X<7)

Answer 1

Let x = number of patients who are female

Given that

sample size = n = 10

population proportion = p = 0.432

A ) we want to calculate the probability that at least 1 patient is a female in the 10 selected, P(X≥1)

i.e.

By using Binomial distribution has p.m.f. is

we want to calculate

i.e.

  

For X = 0,

  

Therefor  

  

   ................( ANSWER)

==============================================================================

B ) we want to calculate probability that more than 3 but less than 7 patients are female, P(3<X<7)

i.e.

By using Excel we can calculate the this probabilities as

Excel command is =BINOMDIST(number_s,trials,probability_s,cumulative)

For x = 4   =BINOMDIST(4,10,0.432,FALSE) = 0.2456

For x = 5 =BINOMDIST(5,10,0.432,FALSE) = 0.2242

For x = 6 =BINOMDIST(6,10,0.432,FALSE) = 0.1421

Therefor

  

....................( ANSWER)

Therefor, probability that more than 3 but less than 7 patients are female, P(3<X<7) = 0.6118 ...........( ANSWER)